Consider the linear transformation f:R3→R2 defined by
f(x,y,z)=(−4x+8y,6z−6y).
If B={(1,1,1),(0,1,1),(0,0,1)} is a basis for the domain of f and C={(1,1),(1,−1)} is a basis for the codomain of f then find the matrix [f]CB of f with respect to the basis B and the basis C.

We are asked to find the matrix representation of the linear transformation \( f: \mathbb{R}^3 \to \mathbb{R}^2 \) with respect to the basis \( B \) for the domain and the basis \( C \) for the codomain.

### Step 1: Express the linear transformation
The transformation \( f \) is defined by:

\[
f(x, y, z) = (-4x + 8y, 6z - 6y)
\]

### Step 2: Apply \( f \) to the basis vectors of \( B \)
The basis \( B = \{ (1, 1, 1), (0, 1, 1), (0, 0, 1) \} \) is given for the domain. We will compute the image of each of these vectors under \( f \).

- For \( (1, 1, 1) \):
  \[
  f(1, 1, 1) = (-4(1) + 8(1), 6(1) - 6(1)) = (-4 + 8, 6 - 6) = (4, 0)
  \]

- For \( (0, 1, 1) \):
  \[
  f(0, 1, 1) = (-4(0) + 8(1), 6(1) - 6(1)) = (8, 0)
  \]

- For \( (0, 0, 1) \):
  \[
  f(0, 0, 1) = (-4(0) + 8(0), 6(1) - 6(0)) = (0, 6)
  \]

Thus, we have:
\[
f(1, 1, 1) = (4, 0), \quad f(0, 1, 1) = (8, 0), \quad f(0, 0, 1) = (0, 6)
\]

### Step 3: Express the results in terms of the codomain basis \( C \)
The codomain basis is \( C = \{ (1, 1), (1, -1) \} \). To find the coordinates of the transformed vectors in this basis, we need to express each result as a linear combination of \( (1, 1) \) and \( (1, -1) \).

Let’s represent each image vector as \( a(1, 1) + b(1, -1) \), then solve for \( a \) and \( b \).

- For \( f(1, 1, 1) = (4, 0) \):

\[
a(1, 1) + b(1, -1) = (4, 0)
\]
This gives the system:
\[
a + b = 4
\]
\[
a - b = 0
\]
Adding the two equations, \( 2a = 4 \), so \( a = 2 \). Substituting into \( a + b = 4 \), we get \( b = 2 \).

Thus, \( f(1, 1, 1) = 2(1, 1) + 2(1, -1) \), so the coordinates of \( f(1, 1, 1) \) in the basis \( C \) are \( (2, 2) \).

- For \( f(0, 1, 1) = (8, 0) \):

\[
a(1, 1) + b(1, -1) = (8, 0)
\]
This gives the system:
\[
a + b = 8
\]
\[
a - b = 0
\]
Adding the two equations, \( 2a = 8 \), so \( a = 4 \). Substituting into \( a + b = 8 \), we get \( b = 4 \).

Thus, \( f(0, 1, 1) = 4(1, 1) + 4(1, -1) \), so the coordinates of \( f(0, 1, 1) \) in the basis \( C \) are \( (4, 4) \).

- For \( f(0, 0, 1) = (0, 6) \):

\[
a(1, 1) + b(1, -1) = (0, 6)
\]
This gives the system:
\[
a + b = 0
\]
\[
a - b = 6
\]
Adding the two equations, \( 2a = 6 \), so \( a = 3 \). Substituting into \( a + b = 0 \), we get \( b = -3 \).

Thus, \( f(0, 0, 1) = 3(1, 1) - 3(1, -1) \), so the coordinates of \( f(0, 0, 1) \) in the basis \( C \) are \( (3, -3) \).

### Step 4: Construct the matrix \( [f]_{C \leftarrow B} \)
The matrix \( [f]_{C \leftarrow B} \) consists of the coordinates of \( f(1, 1, 1) \), \( f(0, 1, 1) \), and \( f(0, 0, 1) \) with respect to the basis \( C \), arranged as columns. Therefore:

\[
[f]_{C \leftarrow B} = \begin{pmatrix} 2 & 4 & 3 \\ 2 & 4 & -3 \end{pmatrix}
\]

This is the matrix representation of the transformation \( f \) with respect to the basis \( B \) in the domain and the basis \( C \) in the codomain.

### Do you have any questions or would you like more details?

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Here are some related questions:
1. What are the properties of a linear transformation matrix between different bases?
2. How would the matrix change if the basis \( B \) or \( C \) were different?
3. What is the geometric interpretation of a transformation from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \)?
4. How do you compute the inverse of a matrix in a different basis?
5. Can you describe the relationship between linear independence and matrix transformations?

**Tip:** When changing bases, it is essential to compute the coordinates of vectors in the new basis. This skill helps in many linear algebra applications like diagonalization and change of basis matrices.

Consider the linear transformation f: R^3 → R^2 defined by f(x,y,z) = (−4x+8y, 6z−6y). If B = {(1,1,1), (0,1,1), (0,0,1)} is a basis for the domain of f and C = {(1,1), (1,−1)} is a basis for the codomain of f, then find the matrix [f]CB of f with respect to the basis B and the basis C.

Learn how to find the matrix representation of the linear transformation f(x, y, z) = (-4x + 8y, 6z - 6y) from R^3 to R^2 with respect to bases B = {(1,1,1), (0,1,1), (0,0,1)} and C = {(1,1), (1,-1)}. This solution includes key linear algebra concepts like basis change and matrix representation. Suitable for undergraduate students in a Linear Algebra course.

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